Given the recursive sequence $\{a_n\}$ defined by setting $0 < a_1 < 1, \; a_{n+1} = a_n(1-a_n) , \; n \ge 1 $
Calculate : $$\lim_{n \rightarrow \infty } \frac{n (1- na_n)}{\log n} $$
My attempts : $$\lim_{n \rightarrow \infty } \frac{n (1- n a_n)}{\log n} =\lim_{n \rightarrow \infty }\frac {n \left (\frac{1}{n a_n} -1 \right) n a_n} {\log n}= \lim_{n \rightarrow \infty } \frac {\frac{1}{a_n} – n}{ \log n}$$
Now I am not able to proceed further.
Please help me.
Thank You.
Best Answer
Note. The answer below may be excessive compared to what OP is asking.
For a quick answer, read the definition of $(x_n)$, jump directly to the proof of proposition, and then read only the first 3 steps.
Let $a_1 \in (0, 1)$ and define $x_n = 1/a_n$. Then $x_n$ solves the following recurrence relation
$$ x_{n+1} = x_n + 1 + \frac{1}{x_n} + \frac{1}{x_n(x_n - 1)}. \tag{1}$$
Using this we progressively reveal the asymptotic behavior of $(x_n)$. More precisely, our goal is to prove the following statement.
We defer the proof to the end and analyze the asymptotic behavior of OP's limit first. Plugging the asymptotic expansion of $x_n$, we find that
$$ r_n := \frac{n(1-n a_n)}{\log n} = \frac{n(x_n - n)}{x_n \log n} = 1 + \frac{C(x_1)}{\log n} + \mathcal{O}\left(\frac{\log n}{n}\right). $$
This tells that, not only that $r_n \to 1$ as $n\to\infty$, but also that the convergence is extremely slow due to the term $C/\log n$.
For instance, $f^{\circ 94}(2) \approx 100.37$ tells that $C(100) \approx C(2) + 94$. Indeed, a numerical simulation using $n = 10^6$ shows that
\begin{align*} x_1 = 2 &\quad \Rightarrow \quad (r_n - 1)\log n \approx 0.767795, \\ x_1 = 100 &\quad \Rightarrow \quad (r_n - 1)\log n \approx 94.3883, \end{align*}
which loosely matches the prediction above.
Proof of Proposition.
Step 1. Since $x_{n+1} \geq x_n + 1$, it follows that $x_n \geq n + \mathcal{O}(1)$. In particular, $x_n \to \infty$ as $n\to\infty$.
Step 2. Since $x_{n+1} - x_n \to 1$, we have $\frac{x_n}{n} \to 1$ by Stolz-Cesaro theorem.
Step 3. Using the previous step, we find that
$$ \frac{x_{n+1} - x_n - 1}{\log(n+1) - \log n} = \frac{1}{(x_n - 1)\log\left(1+\frac{1}{n}\right)} \xrightarrow[n\to\infty]{} 1 $$
So, again by Stolz-Cesaro theorem, we have $x_{n+1} = n + (1+o(1))\log n$. This is already enough to conclude that OP's limit is $1$.
Step 4. By the previous step, we find that $ x_{n+1} - x_n = 1 + \frac{1}{n} + \mathcal{O}\left(\frac{\log n}{n^2}\right)$. Using this, define $C$ by the following convergent series
$$ C(x_1) = x_1 - 1 + \sum_{n=1}^{\infty} \underbrace{ \left( x_{n+1} - x_n - 1 - \log\left(1+\frac{1}{n}\right) \right) }_{=\mathcal{O}(\log n/n^2)}. $$
Splitting the sum for $n < N$ and $n \geq N$ and using the estimate $\sum_{n\geq N}\frac{\log n}{n^2} = \mathcal{O}\left(\frac{\log n}{n}\right)$,
$$ C(x_1) = x_N - N - \log N + \mathcal{O}\left(\frac{\log N}{N}\right), $$
which confirms the first assertion of the proposition.
Once this is established, then the second assertion easily follows by interpreting $x_{n+1}$ as the $n$-th term of the sequence that solves $\text{(1)}$ with the initial value $f(x_1)$. Hence comparing both
$$ x_{n+1} = n+1 + \log(n+1) + C(x_1) + o(1) $$
and
$$ x_{n+1} = n + \log n + C(f(x_1)) + o(1) $$
the second assertion follows. ////